Let \(f\)  and \(f^{\prime }\)  be two asymptotically rigid diffeomorphisms which are properly isotopic; without loss of generality, we may assume that \(f^{\prime }\)  is the identity.
Therefore, there exists some suited subsurface \(M\subset \mathcal {C}_{d,r}(O,Y)\)  such that \(f(M) = M\)  and \(f\)  is the identity when restricted to the complement of \(M\) . By the Alexander Method [1]}, the restriction of \(f\)  to \(M\)  is isotopic, via an isotopy defined on \(M\) , to the identity map on \(M\) . We may extend this isotopy to the complement of \(M\)  by the identity map; this gives the desired proper isotopy between \(f\)  and the identity.
